A consistent modified Zerilli-Armstrong flow stressmodel for BCC and FCC metals for elevatedtemperatures

F. H. Abed and G. Z. Voyiadjis, Baton Rouge, Louisiana

Received July 29, 2004; revised October 27, 2004Published online: February 24, 2005 � Springer-Verlag 2005

Summary. The Zerilli-Armstrong (Z-A) physical based relations that are used in polycrystalline metals at

low and high strain rates and temperatures are investigated in this work. Despite the physical bases used in

the derivation process, the Z-A model exhibits certain inconsistencies and predicts inaccurate results when

applied to high temperatures-related problems. In the Z-A model, the thermal stress component vanishes

only when T!1. This contradicts the thermal activation mechanism that imposes an athermal behavior

for the flow stress at certain finite critical temperatures. These inconsistencies, in fact, are attributed to

certain assumptions used in the Z-A model formulation that causes the model parameters to be inaccu-

rately related to the microstructural physical quantities. New relations are, therefore, suggested and

proposed in this work using the same physical bases after overcoming any inappropriate assumptions. The

proposed modified relations along with the Z-A relations are evaluated using the experimental results for

different bcc and fcc metals. Comparisons are also made with the available experimental results over a

wide range of temperatures and strain rates. The proposed model simulations, in general, show better

correlation than the Z-A model particularly at temperatures values above 300K�. Numerical identification

for the physical quantities used in the definition of the proposed model parameters is also presented.

1 Introduction

Large deformation problems, such as high speed machining, impact, and various primarily

metal forming operations, require constitutive models that are widely applicable and capable of

accounting for complex path of deformation, temperature, and strain rate. The degree of

success of any model mainly depends on: (i) the physical basis used in the derivation process

producing material parameters that are related directly to the nano-/micro-physical quantities;

(ii) the flexibility and simplicity of determining material constants from a limited set of

experimental data; (iii) capturing the important aspects of static and/or dynamic behavior

besides being mathematically and computationally accurate. In dynamic problems that intro-

duce high strain rates, the dynamic yield stress is considered the most important expression

needed to characterize the material behavior and is also used in finite element codes.

In this regard, the dislocation-mechanics-based constitutive relation for material dynamics

calculations developed by Zerilli and Armstrong [1] is considered as one of the most widely used

models that have been implemented in many finite element dynamic codes (ABAQUS, DYNA,

and others) and used by many authors in different types of low and high strain rates and

temperature-related applications (see, for example, [2], [3]). Other authors [4]–[6] reviewed and

Acta Mechanica 175, 1–18 (2005)

DOI 10.1007/s00707-004-0203-1

Acta MechanicaPrinted in Austria

evaluated the predicted inaccuracies as well as the inconsistencies of the Z-A model when

compared to experimental results for different bcc and fcc metals. Yet, no one investigated the

impact on the model due to these inconsistencies as well as the model physical basis. The Z-A

model incorporates strain, strain rate and temperature dependence in a coupled manner. This

model is used in high rates of loading based computer codes. In the Z-A model, the concept of

thermal activation analysis for overcoming local obstacles to dislocation motion as well as the

dislocations interaction mechanisms are used in deriving two different relations for two dif-

ferent classes of metal crystal structures; body centered cubic (bcc) and face centered cubic (fcc).

The differences between the two forms mainly ascribe to the dislocation characteristics for each

particular structure. Fcc metals show stronger dependence of the plastic strain hardening on

temperature and strain rate. Such effect, however, is mainly captured by the yield stress in most

bcc metals. In other words, the thermal flow stress component, which has the coupling effect of

both temperature and strain rate, pertains mainly to the yield stress in bcc metals and to the

hardening stress in fcc metals. That is to say, the cutting of dislocation forests is the principal

mechanism in fcc metals and the overcoming of Peierls-Nabarro barriers is the principal

mechanism in bcc metals.

In spite of the Z-A model physical basis, it is found that the definition of the material

parameters as related to the microstructure physical quantities is inaccurate. These material

parameters lose their physical meaning when used for high temperature and strain rate

applications. This is mainly attributed to the use of certain mathematical expansions in the

derivation of the model, as will be discussed later, in simplifying the physical relations of the

model parameters. Furthermore, the assumption of using an exponential function in describing

the coupling effects of temperature and strain rate on the flow stress produces another

inconsistency to the model in spite of the good experimental fitting for some cases of loading.

The Z-A model in its current form is not able to capture the athermal temperature, which is a

critical value that defines the vanishing stage of the thermal stress. At this critical temperature,

the plastic flow stress pertains totally to the athermal component.

The objective of this paper is to investigate the physical basis of the Z-A constitutive relations

and to modify the physical interpretation of the model parameters by introducing validated

constitutive relations with material parameters that are related accurately to the nano-/micro-

physical quantities. Since the proposed model in this work follows the same physical basis as

the Z-A model, a detailed discussion about the physical basis as well as the procedure used in

deriving the Z-A relations is given in Sect. 2. In Sect. 3, the physical interpretation of the Z-A

model parameters is investigated, and consequently modified relations are proposed. Appli-

cations of the proposed modified model for different metals at low and high strain rates and

temperatures are given in Sect. 4. Comparisons are also made of the proposed model with both

the Z-A model and the experimental results.

2 Physical basis of the Z-A model

The Z-A model is basically derived based on dislocation mechanisms which in fact play a main

role in determining the inelastic behavior of a metal and its flow stress under different load

conditions. The derivation of the aforementioned dislocation basis model uses Orowan’s

equation [7] that defines the dislocation movement mechanisms by relating the equivalent

plastic strain rate _ep ¼ ð2_ep

ij_ep

ij=3Þ1=2 to the density of the mobile dislocations qm, dislocation

speed m, and the magnitude of the Burgers vector b as follows:

2 F. H. Abed and G. Z. Voyiadjis

_ep ¼ ~m b qm m; ð1Þ

where ~m ¼ 2MijMij=3ð Þ1=2 can be interpreted as the Schmidt orientation factor and Mij is the

average Schmidt orientation tensor which relates the plastic strain rate tensor _eij at the

macroscale to the plastic shear strain rate _cp at the microscale as follows:

_eij ¼ _cpMij ¼_cp

2ni � mj þ mi � njð Þ; ð2Þ

where n and m denote the unit normal on the slip plane and the unit vector in the slip direction,

respectively. The average dislocation velocity m can be determined through thermal activation

by overcoming local obstacles to dislocation motion. Many authors have introduced velocity

expressions for thermally activated dislocation glides (see, for more details, [8]–[11]). In this

regard, the following general expression is utilized:

m ¼ mo exp �G=kTð Þ; ð3Þ

where mo ¼ dwo is the reference dislocation velocity, d is the average distance the dislocation

moves between the obstacles, wo is a frequency factor which represents the reciprocal of the

waiting time tw needed for the dislocation to overcome an obstacle,G is the shear stress-dependent

free energy of activation which may depend not only on stress but also on temperature and the

internal structure, k is Boltzmann’s constant, and T is the absolute temperature.

The activation energy G is expressed in terms of the thermal component of the shear stress sth

such that

G ¼ Go �Zsth

0

V �ds0th; ð4Þ

where Go is the reference Gibbs energy at T ¼ 0 and V� is the activation energy volume which is

widely alluded to in the literature as a measure of the Burgers vector b times the area A� swept

out by dislocations during the process of thermal activation. At fixed strain rate, the thermal

stress diminishes and approaches zero as the temperature increases and reaches critical values.

In consequence, the required activation energy G increases until it approaches its maximum

value, Go, as implied by Eq. (4). A mean value of the activation volume V may be used to

characterize the thermal activation process such that [1]

V ¼ V �h i ¼ 1

sth

� �Zsth

0

V �ds0th: ð5Þ

Kocks et al. [12] introduced an empirical definition for the activation energy related to the

thermal shear stress as follows:

sth ¼ s 1� ðG=GoÞ1=q� �1=p

; ð6Þ

where s is the threshold shear stress at which the dislocations can overcome the barriers without

the assistance of thermal activation. The exponents p and q are constants defining the shape

of the short-range barrier where their values are within the ranges (0.0<p �1.0) and (1.0<q �2.0). According to Kocks [13], the typical value of the constant q is 2 that is equivalent to a

triangle obstacle profile near the top whereas the typical value of the constant p is 1/2, which

characterizes the tail of the obstacle.

The thermal flow stress component, for metals in general, is determined, after making use of

Eqs. (1)–(5), as follows:

A consistent modified Zerilli-Armstrong flow stress model 3

rth ¼ c1Ao

A1� ðkT=GoÞ lnð_ep=_epoÞð Þ: ð7Þ

Alternatively, Zerilli and Armstrong expressed the thermal flow stress in Eq. (7) using an

exponential form in order to capture the experimental observation of some metals:

rth ¼ c1e�cT ; ð8Þ

where the physical material parameter c and the threshold stress c1 (stress at zero Kelvin

temperature) are defined as follows:

c1 ¼mGo

Aob; ð9Þ

c ¼ ð1=TÞ lnðA=AoÞ � lnð1þ ðkT=GoÞ lnð_ep=_epoÞ½ �; ð10Þ

where

_epo ¼ ~m bqmmo: ð11Þ

In the above relation, Ao is the dislocation activation area at T ¼ 0, m is the Taylor factor that

relates the shear stress to the normal stress (r ¼ ms), and _epo is the reference equivalent plastic

strain rate which represents the highest strain rate value as it is related to the reference dislo-

cation velocity. It should be noticed here that Eq. (7) could be recovered simply by substituting

Eq. (10) into Eq. (8) using the following mathematical relations: lnða=bÞ ¼� lnðb=aÞ ¼ln a� ln b and a ¼ eln a.

Experimental observations show that the parameter c can be defined as follows [14]:

c ¼ c3 � c4 lnð_ep=_epoÞ: ð12Þ

Hence, the constant material parameters of the Z-A model, c3 and c4, are related to the nano/

micro-physical quantities by comparing Eq. (12) with Eq. (10) such that:

c3 ¼ ð1=TÞ ln A=Aoð Þ; ð13Þ

c4 lnð_ep=_epoÞ ¼ ð1=TÞ ln 1þ ðkT=GoÞ lnð_ep=_epoÞð Þ: ð14Þ

Complying with experimental observations in Eq. (12), Zerilli and Armstrong presumed the

parameter c3 as a constant by considering an implicit temperature dependence of the activation

area A. They also simplified Eq. (14) using the expansion lnð1� xÞ � �x, where

x ¼ ð�kT=GoÞ lnð_ep=_epoÞ, in order to obtain a constant form for the parameter c4 such that:

c4 ¼ k=Go: ð15Þ

The Z-A constitutive relations for bcc and fcc metals are then determined utilizing the concept

of additive decomposition of the flow stress (r ¼ 3rijrij=2ð Þ1=2 for the von Mises case), into

thermal rth and athermal rath components:

r ¼ rth þ rath: ð16Þ

The final form of the thermal and athermal flow stress relations, however, differs from one

metal to another depending on their crystal structure. In turn, two different relations for two

different types of metals were presented; body centered cubic (bcc) and face centered cubic (fcc).

2.1 Z-A relation for bcc metals

The behavior of most bcc metals such as pure iron, tantalum, molybdenum, and niobium shows

a strong dependence of the thermal yield stress on the strain rate and temperature. Moreover,

4 F. H. Abed and G. Z. Voyiadjis

the activation volume V (and in turn, the activation area A since V ¼ Ab where b is constant) is

essentially independent of the plastic strain [14]. In other words, the plastic hardening of most

bcc metals is hardly influenced by both strain rate and temperature. The thermal stress is then

interpreted physically as the resistance of the dislocation motion by the Peierls barriers (short-

range barriers) provided by the lattice itself. Thus, the thermal stress of the Z-A model is

written from Eqs. (8)–(12) as follows:

rth ¼ c1 exp �c3T þ c4T lnð_ep=_epoÞð Þ; ð17Þ

where the material parameters c1, c3, and c4 are related to the microstructure physical quan-

tities as given by Eq. (9), Eq. (13), and Eq. (15), respectively.

On the other hand, the athermal component of the plastic flow stress for bcc metals is mainly

due to the hardening stress that is evaluated from an assumed power law c5enp . In addition, an

extra stress c6, which contributes to the athermal component of the flow stress, is attributed to

the influence of solutes and the original dislocation density as well as the grain size effect. Thus,

rath can be defined as follows:

rath ¼ c5enp þ c6: ð18Þ

Utilizing Eq. (16) together with the definitions given in Eq. (17) and Eq. (18), the final form of

the Z-A constitutive relation for bcc metals is obtained as follows:

r ¼ c1 exp �c3T þ c4T lnð_ep=_ep0Þð Þ þ c5enp þ c6: ð19Þ

Equation (19) clearly shows the uncoupling between the plastic strain hardening and the effect

of thermal softening and plastic strain-rate hardening for most bcc metals.

2.2 Z-A relation for fcc metals

Unlike the case for bcc metals, the thermal activation analysis for most fcc metals like copper

and aluminum is strongly dependent on the plastic strain. This, actually, is attributed to the

thermal activation energy mechanism which is controlled and dominated by the emergence and

evolution of a heterogeneous microstructure of dislocations as well as the long-range inter-

sections between dislocations. In this case, the activation area is related to the plastic strain as

follows:

Ao ¼ A0oe�1=2p : ð20Þ

In turn, the Z-A thermal flow stress for the case of fcc metals is related to the plastic strain,

plastic strain-rate and temperature such that

rth ¼ c2e1=2p exp �c3T þ c4T lnð _ep= _epoÞð Þ; ð21Þ

where the material constant c2 is defined as

c2 ¼ mGo=A0ob: ð22Þ

The thermal component of the constitutive equation for the case of fcc metals clearly shows the

coupling of temperature, strain rate, as well as the plastic strain. The athermal component c6,

however, is constant and independent of the plastic strain and it pertains totally to the initial

yield stress, i.e., to the influence of solutes and the original dislocation density. The Z-A

dislocation–mechanics-based constitutive relation for fcc metals is then written as follows:

r ¼ c2e1=2p exp �c3T þ c4T lnð _ep= _epoÞð Þ þ c6: ð23Þ

A consistent modified Zerilli-Armstrong flow stress model 5

Although the number of material constants in the fcc relation is less than in the corre-

sponding bcc relation, the procedure steps for determining these constants are almost the same

in both relations. The constants c3 and c4, however, share the same physical interpretation for

the bcc and fcc relations.

3 Investigation and modification of the Z-A plastic flow relations

In investigating the physical interpretation of the Z-A constitutive relation parameters, two

crucial points are noticed. First, the explicit definition of c3 given by Eq. (13) clearly indicates

that this parameter is not a constant, but rather a temperature dependent parameter. The only

way to keep this parameter (c3) as a constant is when the relation between the activation area A

and the Kelvin temperature T takes an exponential form which is not necessarily the case in

general. Second, the assumption for the expansion given for lnð1� xÞ � �x in obtaining the

parameter c4 in Eq. (15) is not an accurate one since this expansion is valid only for values

x 1:0. The variable x ¼ G=Goð Þ ¼ �kT lnð _ep= _epoÞ=Goð Þ is actually both temperature and

strain rate dependent. The percentage change between the exact and approximated x-values

(initiated at x 0:1) affects significantly the prediction of the flow stresses. This because the x-

values are due to an exponential term that is already multiplied by a large stress number which

is the threshold stress, c1, for the bcc model, in Eq. (19), and the hardening value, c2enp , for the

fcc model, Eq. (23).

As widely defined in the literature (see, for example, [15], [16]), the numerical values of the

reference Gibbs free energy, Go, ranges between 0.6 to 1.0 eV/atom for most bcc metals and 1.5

to 2.5 eV/atom for fcc metals. In addition, the reference plastic strain rate _epo values are of the

order 105 to 107s�1 for most metals whereas the Boltzmann’s constant k value is taken as

8.62�10�5 eV/K (1.3807�10�23 JK)1). Thus, the numerical investigation of the variable x using

average values of the abovementioned quantities elucidates that x increases and approaches the

value of 1.0 when the temperature increases and the strain rate decreases as shown in Fig. 1. It

is clear that at strain rates up to 10þ3s�1 the numerical values of the x variable exceed 0.1 for

relatively low temperatures (110–350K�). Thus, the Z-A model is physically justified when it is

used at temperature and strain rate combinations that fall in the region zone indicated below

the horizontal dotted line (x ¼ 0.1) shown in Fig. 1. Conversely, at greater x values (in the

region above the aforementioned horizontal dotted line), the physical interpretation of the

parameter c4 given by the Z-A model becomes ineffective and does not reflect the real behavior.

This, in turn, makes the model parameters appear to be more phenomenologically defined than

physically based. Additionally, it is known that the considered mechanism becomes athermal

(thermal stresses vanish) when the activation energy G approaches its reference value Go (i.e.,

x ¼ 1.0) at a critical temperature that is termed the athermal temperature (Tcr). The Z-A model,

in turn, fails to disclose the above behavior since the only way for the thermal stress component

to vanish is when T !1 (due to the assumed exponential dependence of the temperature).

This, in fact, makes the Z-A model physically inconsistent and clearly in contradiction with Eq.

(4) which is used along with Eqs. (1) and (3) as the base of the thermal activation analysis.

Zerilli and Armstrong [1], [17] applied their model to tantalum and OFHC copper as bcc and

fcc metals respectively. They determined the material constants based on experimental results

provided by other authors for low and high strain rates and temperatures. For tantalum, the

material parameters are obtained from the strain-rate dependence of the lower yield stress at

room temperature (T=300K�) such that c4 is equal to 0.000327 and for the case of OFHC

copper is around 0.000115. Comparing the above numerical values of the parameter c4 to its

6 F. H. Abed and G. Z. Voyiadjis

simplified definition given by Eq. (15) and using the constant value of k, the numerical values of

Go are found to be around 0.26 eV/atom for tantalum and 0.75 eV/atom for OFHC copper.

These values, however, are much lower than those obtained in the literature by many authors.

For example, Hoge and Mukharjee [18] who provided the experimental results for tantalum

used a value of 0.62 eV/atom in their modeling that is necessary to nucleate a pair of kinks.

Moreover, Nemat-Nasser and Li [16] employed their experimental results in modeling the

plastic flow of OFHC copper at low and high strain rates and temperatures by using a 1.75 eV/

atom for the physical quantity Go (see also [19]). The deviations in Go between the value given

by the Z-A model and that found in the literature are actually attributed to the adoption of the

expansion lnð1� xÞ � �x which actually directed the model derivation to match the experi-

mental evidence given by Eq. (12).

Furthermore, the Z-A model does not really show an explicit form for the temperature

dependence of the activation area. In contrast, the Z-A model introduced an implicit activation

area dependence of the temperature through the definition of the material constant c3. This

constant value, however, exists only when the exponential relationship mentioned earlier be-

tween the temperature and the activation area is defined as A ¼ Aoec3T which clearly indicates

that the effect of the strain rate on the activation area is not considered in the Z-A model. Such

effect however has been investigated by many authors (see, for more details, [20]). The acti-

vation area behaves predictably in terms of the effects of parameters such as dislocation density

and solid solution concentration. In the classical studies of dislocation plasticity, the activation

area is calculated based on the rate sensitivity of the flow stress measured using rate-change or

stress relaxation experiments as discussed in [21], [22]. The following definition, which depends

on both temperature and strain rate effect, is considered one of the ways for measuring the

activation area A:

A ¼ kT=ðb@s=@ ln _e�pÞ; ð24Þ

where _e�p ¼ _ep= _epo. The above relation is used to calculate the activation volumes from the

strain rate dependence of the flow stress at constant temperature as a function of the effective

stress [17], [18]. These activation volumes decrease with the increase of thermal stresses and are

much smaller than those obtained when the intersection mechanism is operative. It is found

that the activation area for the fcc metal deformation is 10 to 100 times larger than those found

in the deformation of bcc metals which is in the range of 5 to 100b2.

Eventually, in order to have a solid physically based constitutive relation, an explicit defi-

nition is essential for the activation area in terms of temperature and strain rate in order to be

0

0.10.2

0.3

0.40.5

0.6

0.7

0.80.9

1

0 200 400 600 800 1000 1200Temperatute (K)

x -v

alue

0.0001/s0.1/s10/s1000/s10000/s

Fig. 1. Variation of x ¼ ð�kT=G0Þ lnð _ep= _ep0Þ values with temperatures atlow and high strain rates

A consistent modified Zerilli-Armstrong flow stress model 7

used in the definition of the thermal stress given by Eq. (7). To the authors’ knowledge, no such

definition that is physically derived and explicitly related to both temperature and strain rate

currently exists in the literature. At this moment, however, the authors suggest a new definition

for the activation area which is derived utilizing Eq. (24) along with Eq. (6) after employing the

typical values of the exponents p and q mentioned earlier such that:

A ¼ Ao 1� ð�kT=GoÞ ln _e�p

� �1=2� ��1

: ð25Þ

It is obvious from Eq. (25) that the activation area increases as the temperature evolves

throughout the plastic deformation. Conversely, it decreases as the strain rate increases due to

the insufficient time required for dislocations to move inside the lattice. Moreover, A ap-

proaches its reference value Ao at zero Kelvin temperature.

A similar definition for the activation area in Eq. (25) may be obtained by directly consid-

ering the same variation of the thermal stress with the activation energy given in Eq. (6) since

the activation area is proportionally related to the threshold stress denoted by s or by c1 as

given in Eq. (9). This variation may exist at any loading and temperature stages. That is the

stress is expected to be proportionally related to the activation area at any temperature,

r / 1=A, as implied by Eq. (24).

In light of the explicit derived definition for the thermal stress along with the suggested

temperature and strain dependence of the activation area given in Eq. (25), the Z-A constitutive

relations are then redefined such that the physical definitions of the model parameters are

appropriately justified and related to the nano/micro-structure quantities. These two proposed

flow stress relations are defined for both bcc and fcc metals by the following Eqs. (26) and (27),

respectively:

r ¼ c1 1� X1=2 � X þ X3=2� �

þ c5enp þ c6; ð26Þ

r ¼ c2e0:5p 1� X1=2 � X þ X3=2� �

þ c6; ð27Þ

where the variable X ¼ x ¼ ð�kT=GoÞ ln _e�p or X ¼ c4T lnð1= _e�pÞ and the physical parameters ci

are the same as previously defined.

Equations (26) and (27) may be used to predict the stress strain curves for both isothermal

and adiabatic plastic deformations. For the case of adiabatic deformation, heat inside the

material increases as plastic strain increases and therefore the temperature T is calculated

incrementally by assuming that the majority of the plastic work is converted to heat:

T ¼ fcpq

Zep

0

rd ep: ð28Þ

Here q is the material density and cp is the specific heat at constant pressure. The Taylor-

Quinney empirical constant v is often assigned the values 0.9–1.0 (see, for more details, [23]).

Unlike the Z-A model, the proposed modified flow stress relations clearly point out that

the thermal stress component for both bcc and fcc metals, which is given by the first term on the

right-hand side of Eqs. (26) and (27), respectively, vanishes as the variable X approaches

the value of 1 or, in other words, G! Go. This mechanism, however, coincides well with the

thermal activation energy mechanism defined in Eq. (4). Based on that, the athermal or critical

temperature, Tcr, may be calculated by setting X ¼ 1 such that:

Tcr ¼ ð�k=GoÞ lnð _ep= _epoÞð Þ�1¼ c4 lnð1= _e�pÞ� ��1

: ð29Þ

8 F. H. Abed and G. Z. Voyiadjis

The above definition indicates clearly that Tcr is strain rate dependent and increases as the

plastic strain rate increases as will be shown later in the model applications section.

3.1 Further modifications for the proposed model

The plastic hardening for the proposed model, which is evaluated from an assumed power law

dependence mentioned earlier, may alternatively be related to the forest dislocation density

through the dislocation model of Taylor [24]. This gives the shear flow stress s in terms of the

forest dislocation density �qf where �qf ¼ qf � qi and qi denotes the initial dislocation density

encountered in the material due to the manufacture process or by nature,

s ¼ albffiffiffiffiffi�qf

p; ð30Þ

where a is an empirical coefficient taken to be 0.2 for fcc metals and about 0.4 for bcc metals as

given by Nabarro et al. [25]. Nabarro pointed out that Eq. (30) can be derived by multiple

methods. Ashby [26] splits the difference of the a values in assuming that a ¼ 0.3 for most

metals. Kubin and Estrin [27] proposed the following set of two coupled differential equations

to describe both forest qf and mobile qm dislocation density evolutions with plastic strain as

follows:

dqm

dep

¼ k1

b2� k2qm �

k3

bq1=2

f ;

dqf

dep

¼ k2qm þk3

bq1=2

f � k4qf ; ð31Þ

where the constant coefficients ki are related to the multiplication of mobile dislocations (k1),

their mutual annihilation and trapping (k2), their immobilization through interaction with

forest dislocations (k3), and to the dynamic recovery (k4). Klepaczko [28] showed that the

growth of dislocation density is nearly linear with regard to the deformation in the first steps of

the hardening process, independently of the temperature. This is followed by a recombination

of the dislocations that are assumed to be proportional to the probability of dislocation

meeting, that is to say of the forest dislocation density. Based on this hypothesis, the following

simple relation for the evolution of the forest dislocation density qf was presented:

dqf

dep

¼ M � Ka �qf ; ð32Þ

where M is the multiplication factor and ka is the dislocation annihilation factor which may

depend on both temperature and strain rate. Klepaczko and Rezaig [29] showed that for mild

steels both M and ka could be assumed constant at not so high strain rates and up to the

temperature where the annihilation micromechanisms (recovery) start to be more intense. On

this basis, the hardening stress for metals may be expressed in terms of the internal physical

quantities by substituting the plastic strain evolution of the forest dislocation density Eq. (32),

after proper integration, into the Taylor definition Eq. (30) as follows:

Benp � �B 1� expð�kaepÞð Þ1=2; ð33Þ

where �B is the hardening parameter defined as

�B ¼ malb M=kað Þ1=2: ð34Þ

In order to enable one to compare with Z-A model, the above modification for the hardening

stress, Eq. (33), will not be used in the present application. In addition, the following

A consistent modified Zerilli-Armstrong flow stress model 9

comparisons will be mainly focusing on the physical justification of the thermal parameters c3

and c4 as well as on the correlation of both the Z-A and the proposed models with the

experimental results at different strain rates and temperatures.

4 Applications, comparisons, and discussion

The modified constitutive relations derived in the previous section for bcc and fcc metals are

evaluated by direct comparison with the experimental results that are provided by several

authors for different types of metals and conducted over a wide range of strain rates and

temperatures. The evaluation of the material parameters of the proposed model is first

initiated by studying the stress-temperature relation at different values of plastic strain and

at certain strain rate values. Various numerical techniques can be used to determine the

material parameters of the proposed model. However, the Newton-Raphson technique is

applied for both bcc and fcc relations utilizing the available experimental results at different

strain rates and temperatures. The material constants of the proposed relations (Eq. (26)

and Eq. (27)) and the Z-A relations (Eq. (19) and Eq. (23)) are listed in Table 1 for both

OFHC copper and tantalum. Applications as well as comparisons to the available experi-

mental data for OFHC copper (fcc) and tantalum (bcc) using the same material constants

are then illustrated. The proposed model is further applied to other bcc metals such as

vanadium and niobium.

4.1 Application to OFHC copper (Cu)

Oxygen Free High Conductivity (OFHC) copper is used here as an application for the proposed

fcc model to show the temperature and strain rate variation of the flow stress. The material

parameters of the proposed model, Eq. (27), and the Z-A model, Eq. (23), are obtained from

essentially the same data base found in [30]. The predicted flow stresses, generally, agree well

with most experimental data for several strain rates and temperatures. In Fig. 2, the adiabatic

stress-strain calculated using both models predicts good results as compared to the same

experimental data. The modified model, however, shows better correlations with the experi-

mental results at higher temperatures (T=730K). On the other hand, both models underesti-

mate the isothermal stress-strain as compared with the experimental results at room

temperature.

Table 1. Parameters of OFHC Cu and Ta for the proposed and Z-A models

Parameters OFHC Copper Tantalum

Proposed Z-A [1] Proposed Z-A [17]

c1 (Mpa) – – 1125 1125

c2 (Mpa) 970 890 – –

c3 þ k ln _epo=Go – 2.8� 10�3 – 5.35� 10�3

c4 (K�1) 3.55� 10�5 1.15� 10�4 9.37� 10�5 3.27� 10�4

_epo (s�1) 1.76� 108 – 4.45� 106 –

c5 (Mpa) – – 300 310

c6 (Mpa) 50 65 50 30

n – – 0.45 .44

10 F. H. Abed and G. Z. Voyiadjis

A further assessment of the proposed model as well as the Z-A model is made, using the same

material constant, by comparing the adiabatic and isothermal stress-strain results to other

experimental data presented in [16], [31] as shown in Figs. 3 and 4, respectively. Results

obtained using the proposed model compare well with both experimental data at low and high

temperatures (77–1000K�) and the indicated high strain rates (4000 s�1 and 6000 s�1). The

adiabatic flow stresses computed by the Z-A model, nevertheless, fail to match the experimental

results at temperatures greater than 300K�. These predictions, as a point of fact, underestimate

the experimental results at high temperatures due to the exponential stress-temperature rela-

tionship assumed in the derivation process of the Z-A model.

For OFHC copper, the adiabatic stress strain curves predicted by the proposed model are

obtained based on the assumption of conversion of 90% of the plastic work using the formula

defined in Eq. (28). Moreover, the specific heat and the material densities are chosen to be

0.383J/g.K� and 8.96g/cm3, respectively.

4.2 Application to tantalum (Ta)

Similar to the Z-A model, the experimental data presented in [18] are utilized here in deter-

mining the material constants for the proposed model. The temperature variation of the flow

stress obtained experimentally by Hoge and Mukharjee [18] at 0.014 strain and 0.0001s�1 strain

rate is compared with the results computed using both the present model Eq. (26) and the Z-A

0

100

200

300

400

500

0 0.25 0.5 0.75 1 1.25 1.5Plastic strain

Stre

ss (

MPa

)

Proposed modelZ-A modelAdiabatic (451 /s, To=294 K)Isothermal (0.002 /s, T=298 K)Adiabatic (464 /s, To=730 K)

Fig. 2. Adiabatic and isothermalstress-strain curves for the proposed

model, for OFHC copper, as com-pared to experimental results [1] and

the Z-A model at various strain ratesand temperatures

0

100

200

300

400

500

600

700

800

0.00 0.20 0.40 0.60 0.80 1.00 1.20True strain

Tru

e st

ress

(M

Pa)

Proposed modelZ-A modelExp. (strain rate = 4000/s) To = 77K

296 K

596 K

896 K

1096K Fig. 3. Adiabatic stress-strain curves

for the proposed model, for OFHCcopper, as compared to experimental

results [16] and the Z-A model at4000s)1 strain rate and different

initial temperatures

A consistent modified Zerilli-Armstrong flow stress model 11

model Eq. (19). The proposed model as well as the Z-A model predicts results that agree very

well with experimental data as shown in Fig. 5. The stress variation with strain rates at room

temperature computed using both models is compared with the same experimental data. The

comparisons illustrated in Fig. 6. show very good agreement with the experimental results.

The results of both model simulations are also compared with another set of experimental data

[15] at 5000s�1 strain rates and elevated temperatures (up to 798K�) as shown in Fig. 7. The

adiabatic stress-strain relations predicted by the proposedmodel show very good agreement at all

indicated temperatures whereas the Z-A model fails again to correlate well with another set of

experimental results particularly at higher temperatures in spite of the good agreement with the

experimental data used in obtaining the model constants. This, in reality, is mainly attributed to

the fitted hardening parameters which are affected indirectly by the exponential stress-tempera-

ture relationship used in the Z-A model for modeling the thermal yield stresses.

In the case of bcc metals, the temperature and strain rate effects emerge clearly at the starting

point (initial yielding) of the stress-strain curves whereas the hardening stress is independent of

both temperature and strain rate effects. Therefore, the predicted stress strain curves using the

proposed and Z-A models will depend completely on the accuracy of determining the hardening

parameters. Moreover, the hardening parameters control the adiabatic stress strain curves by

assuming 90% of the plastic work is converted into heat where the temperature evolution is

calculated using Eq. (28). The specific heat and the material density values for tantalum are

chosen to be 0.139J/g.K� and 16.62g/cm3, respectively.

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8 1Plastic strain

Stre

ss (

MPa

)

Proposed modelZ-A modelAdiabatic (6000/s, To=298K)Isothermal (0.0004/s, T=298K)

Fig. 4. Adiabatic and isothermalstress-strain curves for the proposed

model, for OFHC copper, as com-pared to experimental results [31]

and the Z-A model at different strainrates and room temperature

0

100

200

300

400

500

600

700

800

0 100 200 300 400 500Temperature (K)

Stre

ss (

Mpa

)

Proposed modelZ-A modelExp. (strain rate = 0.0001/s)

Fig. 5. Stress-temperature results forthe proposed and Z-A models, for

tantalum (Ta), as compared withexperimental data [18] at 0.0001 s�1

strain rates

12 F. H. Abed and G. Z. Voyiadjis

4.3 Discussion

It is obvious from the proposed model as well as the Z-A model that the thermal component of

the flow stress is nearly the same for both bcc and fcc metals. However, the model parameters

values and accordingly the physical quantities differ from metal to metal. The physical quan-

tities for OFHC copper and tantalum may be defined after making use of the numerical values

of the material constants defined in Table 1. For the proposed model, it is found that the Gibbs

free energy for Cu (Go � 2.4 eV/atom) is higher than the value assigned for tantalum (Go � 0.9

eV/atom). This, in fact, is due to the two different thermal activation mechanisms for each

metal structure which indicates that the dislocation interaction mechanism necessitates higher

activation energy than the Peierls mechanism in overcoming the short-range barriers. The

numerical values of Go, however, fall in the same range used in the literature that is mentioned

in the previous section.

In general, most metals contain an initial amount of dislocations which are naturally excited

or generated through the manufacturing process. These dislocation densities, however, help

metals deform plastically until a level where no further dislocation generation is allowed which

indicates that the saturation limit of dislocation densities is reached. The initial and saturated

values of the dislocation densities, however, differ from metal to metal (see, for more details,

[19]). In this work, the mobile dislocation density evolution with the plastic accumulations is

not considered and rather an average value of the mobile dislocation density is assumed for

both models.

0

100

200

300

400

500

600

700

800

0.00001 0.001 0.1 10 1000 100000Strain rate (1/s)

Stre

ss (

Mpa

)

Proposed modelZ-A modelExp. (T=298K)

Fig. 6. Stress versus strain rates

results for the proposed and Z-Amodels, for tantalum, as compared to

experimental data [18] at room tem-perature

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Plastic strain

Stre

ss (

Mpa

)

Proposed modelZ-A modelExp. (strain rate = 5000/s)

To = 298K

396K

796K596K496K

Fig. 7. Adiabatic stress-strain curves

for the proposed model, for tantalum,as compared to experimental data [15]

and the Z-A model at 5000s�1 strainrate and different initial temperatures

A consistent modified Zerilli-Armstrong flow stress model 13

The reference plastic strain rate values obtained for both OFHC copper and tantalum,

listed in Table 1, are justified by adopting reasonable quantities for the nano/micro-physical

parameters. On this basis, the average mobile dislocation densities are found to be in the

order of 1014 m�2 for OFHC Cu and 1013 for Ta. The rational for the differences of qm

values in the two materials mainly depends on the dislocation characteristics and interaction

mechanisms for each particular structure. It should be noted also that these values are a

portion of the total dislocation densities which in addition include the forest dislocation

densities introduced through the hardening parameters. The numerical quantities of the

reference velocity (mo ¼ d=tw), on the other hand, show lower values for OFHC Cu O(103s�1)

than Ta O(102s�1) since the waiting time tw needed by the dislocation to overcome an

obstacle for the case of fcc metals is higher than the time required in bcc metals [12]. The

average values of the Burgers vector are taken as 2.5A and 2.9A for OFHC copper and

tantalum, respectively.

The material constants of the Z-A model for both OFHC Cu and Ta found in [1], [17] show

some difficulty in trying to relate their numerical values to the nano/micro physical quantities

particularly for the interpretation of the (c3 þ k ln _epo=Go) constant values listed in Table 1.

This, as discussed in Sect. 3, is due to the improper definition of c3 that is related to the

activation area A which is, in fact, a temperature and strain rate dependent parameter as

explained in the previous section. Finally, one should admit here that the procedure followed in

determining the material constants is found easier for the case of the Z-A model than the one

used for the proposed modified model. This, in fact, is at the cost of the accuracy of the physical

interpretation for the material parameters.

4.4 Application to other bcc metals

The proposed model is further verified by comparing the adiabatic stress strain results predicted

by Eq. (26) to the experimental data of other bcc metals such as vanadium (V) and niobium

(Nb) conducted at high strain rates and for a wide range of temperatures [32], [33]. The material

constants for V and Nb are determined using the same procedure followed in Ta and listed in

Table 2.

The comparisons between the proposed model and the experimental results for the three

indicated metals show very good agreement for most loading conditions as displayed in

Figs. 8 and 9. For all of these three metals, a conversion of 100% of the plastic work is

considered in computing the adiabatic stress strain curves. The incremental evolution of the

absolute Kelvin temperature is obtained, after making use of Eq. (28), by employing the

specific heat values of 0.498J/g.K� and 0.265J/g.K� and the material density values of 6.16g/

cm3 and 8.57g/cm3 for both vanadium and niobium, respectively. It should be mentioned

here that the damage mechanism is not considered in this work and therefore the softening

Table 2. Parameters of V and Nb for the proposed model

Parameters Vanadium Niobium

c1 (Mpa) 945 945

c4 (K�1) 1.392 � 10�4 1.49 � 10�4

_epo (s�1) 6.32 � 106 7.07 � 106

c5 (Mpa) 305 440

c6 (Mpa) 60 60

n 0.16 0.25

14 F. H. Abed and G. Z. Voyiadjis

behavior captured by the proposed modified models are due to the adiabatic deformation

from which the temperature evolves inside the material with the accumulation of the plastic

work.

4.5 Evaluation of the critical temperature Tcr

Ignoring the athermal behavior of the thermal activation mechanism for certain temperatures is

considered one of the major inconsistencies that the Z-A model suffers due to the assumptions

mentioned earlier. The proposed model, on the other hand, is able to introduce this behavior

through a strain rate dependent relation of the critical temperature Tcr as defined in Eq. (34). In

fact, Tcr is defined as the highest temperature value that corresponds to the minimum thermal

stresses (zero thermal stress) observed during the degradation process of the flow stress as the

material temperature increases. Figure 10 shows the strain rate variation of Tcr calculated using

Eq. (34) for the four metals used in this study. It is found that Tcr values increase with in strain

rates increase. Also, the critical temperature value for OFHC Cu is higher than those obtained

for the three bcc metals. This disparity may be attributed to the variation of the reference

plastic strain rate _epo values and also to the different c4 values which are mainly related to the

Gibbs free energy Go.

0

200

400

600

0.00 0.10 0.20 0.30 0.40 0.50 0.60True strain

Tru

e st

ress

(M

Pa)

Proposed modelExp. (strain rate = 2,500 /s)

To = 296K

400K 500K 600K

800K

Fig. 8. Adiabatic stress-strain curvesfor the proposed model, for vana-

dium, as compared to experimentaldata [32] and the Z-A model at

2500s�1 strain rate and differentinitial temperatures

0

200

400

600

800

1000

1200

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70True strain

Tru

e st

ress

(M

Pa)

Present modelExp. (strain rate = 3300/s)

To = 77K190K296K400K

700K 600K

Fig. 9. Adiabatic stress-strain curves

for the proposedmodel, forNiobium,as compared to experimental data

[33] and the Z-A model at 3300s�1

strain rate and different initial tem-

peratures

A consistent modified Zerilli-Armstrong flow stress model 15

The variation of Tcr results within the three bcc metals is mainly ascribed to the variation of Go

values since the _epo values are approximately of the same order. The higher the reference

activation energy values (0.90, 0.62, 0.58 eV/atom for Ta, V, and Nb, respectively) needed to

overcome the barriers the higher are the critical temperature values achieved. Actually, the

thermal stress is interpreted as the resistance of the barriers to the dislocation movement, thus,

the barriers that need higher activation energy to be overcome require also higher temperature

to produce this thermal energy until the resistance of these barriers is completely vanished

which indicates zero thermal stresses.

5 Conclusions

The concept of thermal activation energy along with the dislocation interactions mechanism is

used in this work for modeling the flow stress for both bcc and fcc metals. The Z-A model

assumed an exponential stress-temperature relationship in modeling the thermal stress com-

ponent based on experimental observations. This exponential form is inappropriate for all types

of metals particularly at elevated temperatures. This, in turn, causes the thermal stress com-

ponent of the Z-A model to never vanish at any temperature which is inconsistent with the

considered mechanisms that become athermal when G! Go at certain critical temperatures.

On the other hand, the assumption of using the expansion lnð1þ xÞ � x in obtaining the Z-A

model in its final form is inaccurate for all loading conditions. This assumption causes the

model parameters to be phenomenologically based rather than physically interpreted. Conse-

quently, the Z-A physically based relations for bcc and fcc metals are modified here such that

the material parameters are physically deduced and accurately related to the nano/micro-

structure physical parameters. The nonlinear stress-temperature relationship derived in this

work shows very good correlations with the experimental results for OFHC copper, tantalum,

vanadium, and niobium. Besides, the adiabatic stress-strain relations computed using the

proposed relations show relatively good correlations over wide ranges of temperatures and

strain rates. In contrast, the results predicted by the Z-A model show wide deviation from the

experimental results of the OFHC copper and tantalum particularly at higher temperatures.

Finally, the numerical identification of the physical parameters in the nano/micro-scale dem-

onstrates reasonable quantities as compared to those specified in the literature.

0

500

1000

1500

2000

2500

3000

3500

4000

0.001 0.1 10 1000Strain rate (1/s)

Cri

tical

tem

pera

ture

Tcr

(K

)

OFHC Cu

NbV

Ta

Fig. 10. Strain rate variation of the

critical temperatures predicted usingthe proposed model for different types

of metals

16 F. H. Abed and G. Z. Voyiadjis

Acknowledgement

The authors acknowledge the financial support under grant no. M67854-03-M-6040 provided by the

Marine Corps Systems Command, AFSS PGD, Quantico, Virginia. They thankfully acknowledge theirappreciation to Howard ‘‘Skip’’ Bayes, Project Director. The authors also acknowledge the financial

support under grant no. F33601-01-P-0343 provided by the Air Force Institute of Technology,WPAFB, Ohio.

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Authors’ address: F. H. Abed and G. Z. Voyiadjis (E-mail: voyiadjis@eng.lsu.edu), Department of Civiland Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

18 F. H. Abed and G. Z. Voyiadjis: A consistent modified Zerilli-Armstrong flow stress model